Some aspects of particle physics are more easily understood if we first express the Dirac equation in a more algebraic form than usual, with the gamma matrices replaced by equivalent operators from vector and quaternion algebra. Here, we define unit quaternion operators (1, a, b, c) according to the usual rules, and also multivariate 4-vector operators (i, x, y, z), which are isomorphic to complex quaternions or Pauli matrices.
The combination of these two sets of units produces a 32-part algebra (or group of order 64, taking into account both + and – signs), which can be directly related to that of the five γ matrices. If we now apply a free-particle solution, such as ψ = Aexp[-i(Et - p•r)], to this equation, we find that:
(zE + ixp + iym) Aexp[-i(Et - p•r)]=0,
where p is a multivariate vector. The equation is only valid when A is a multiple of (zE + ixp + iym). In principle, this means that A, and hence ψ, must be a nilpotent or square root of zero. Here, of course, we rely on the fact, that, for a multivariate p, the product pp becomes identical to the product of the scalar magnitudes pp = p2. It is, additionally, identical to the product of the helicities (σ•p) (σ•p), indicating that the multivariate vector (or equivalent Pauli matrix) representation of p automatically incorporates the concept of spin.